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\section{Traditional Notion of Pseudogroups}

We use the definition from Kobayashi and Nomizu~\cite{kobayashi1963foundations}.

\begin{defn}
A \define{Pseudogroup of Transformations} on a topological space
$S$ is a set $\Gamma$ of transformations satisfying the following
axioms:
\begin{enumerate}
\item Each $f\in\Gamma$ is a homeomorphism of an open set (called
  the domain of $f$) of $S$ onto another open set (called the
  range of $f$) of $S$;
\item If $f\in\Gamma$, then the restriction of $f$ to an
  arbitrary open subset of the domain of $f$ is in $\Gamma$;
\item Let $U=\cup_{i}U_{i}$ where each $U_{i}$ is an open set of
  $S$. A homeomorphism $f$ of $U$ onto an open set of $S$ belongs
  to $\Gamma$ if the restriction of $f$ to $U_{i}$ is in $\Gamma$
  for every $i$;
\item For every open set $U$ of $S$, the identity transformation
  of $U$ is in $\Gamma$;
\item If $f\in\Gamma$, then $f^{-1}\in\Gamma$;
\item If $f\in\Gamma$ is a homeomorphism of $U$ onto $V$ and
  $f'\in\Gamma$ is a homeomorphism of $U'$ onto $V'$ and if
  $V\cap U'$ is non-empty, then the homeomorphism $f'\circ f$ of
  $f^{-1}(V\cap U')$ onto $f'(V\cap U')$ is in $\Gamma$.
\end{enumerate}
\end{defn}

So what's the significance of being a homeomorphism? It means
that topologically, the open subsets of $S$ are ``the same''
locally. That is, we can deform any open subset into any other
open subset.

We have some consistency conditions too. This is precisely the
importance of properties 2 and 3. More precisely these are
conditions specifying ``consistency on restrictions''. The last
condition, property 6, is an explicit condition for ``consistency
on overlaps''. 

The property 4 makes a pseudogroup ``sufficiently nice'';
similarly, property 5 indicates that this is kind of like a
groupoid. However, a pseudogroup is \emph{a single set} of
transformations; a groupoid is more structured than this.
